A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data:
| Number of cars | 0 - 10 | 10 - 20 | 20 - 30 | 30 - 40 | 40 - 50 | 50 - 60 | 60 - 70 | 70 - 80 |
|---|---|---|---|---|---|---|---|---|
| Frequency | 7 | 14 | 13 | 12 | 20 | 11 | 15 | 8 |
The mode of the data is 44.7 (approx).
Explanation
The solution follows the method established in Example 6 of the provided context. The modal class is identified as the interval with the highest frequency (20), which is 40-50. Using the standard formula for grouped data, values for the lower limit (l), class size (h), frequency of the modal class (f1), and frequencies of the preceding (f0) and succeeding (f2) classes are substituted to find the mode.
Solution Steps
Step 1: Identify the modal class. The class with the highest frequency is 40-50 (frequency = 20). Therefore, the modal class is 40-50.
Step 2: Identify the values for the formula. Lower limit (l) = 40 Class size (h) = 10 Frequency of modal class (f₁) = 20 Frequency of class preceding modal class (f₀) = 12 Frequency of class succeeding modal class (f₂) = 11
Step 3: Apply the mode formula. Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
Step 4: Substitute the values. Mode = 40 + [(20 - 12) / (2(20) - 12 - 11)] × 10 Mode = 40 + [8 / (40 - 23)] × 10 Mode = 40 + (8 / 17) × 10
Step 5: Calculate the final result. Mode = 40 + 80/17 Mode = 40 + 4.705... Mode 44.7